In this paper, we use a dual currency Lagos-Wright model to explore the nature of optimal monetary policy under currency competition using different timing protocols. The central banks are utility maximizing players. To characterize equilibrium with reputation, we model the centralized market sub period of the Lagos-Wright economy as an infinitely repeated game between the two Leviathan central banks (long run players) and a continuum of competitive agents (short run players). Concentrating on Markov strategies in such a game shows that the Markov perfect equilibrium features highest inflation tax. However, allowing for reputation concerns improves the inflation outcome. Such a game typically features multiple equilibriums but the competition between the banks allows the use of renegotiation proof-ness as an equilibrium selection mechanism. Accordingly, equilibrium featuring the lowest inflation tax is weakly renegotiation proof, suggesting that better inflation outcome is more likely in the case of Leviathan currency competition than in the single Leviathan bank case.